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UNIVERSITI PUTRA MALAYSIA
CLASSIFICATION OF SECOND CLASS FILIFORM LEIBNIZ ALGEBRAS IN DIMENSION NINE
FATANAH DERAMAN
FS 2013 99
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LEIBNIZ ALGEBRAS IN DIMENSION NINE
By
FATANAH DERAMAN
Thesis Submitted to the School of Graduate Studies,
Universiti Putra Malaysia, in Fulfilment of the Requirements for the
Degree of Master of Science
August 2013
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DEDICATIONS
With Special Love,Deraman Ismail and Rohani Ibrahim
Also goes my beloved,Meriam Hj Jalal
For,Mardziah Deraman and Shukri Abd. Karim
Sara ShukriAriff Fahmi Deraman and Azie Azwa Kamaruddin
Muhammad Naim DeramanSiti Amnah DeramanNurul Aunie DeramanWafi Arawi Deraman
Muhammad Farees DeramanIlham Aqil DeramanLast but not least,
Zahela Ismail and Zulkurnain Ismail For all their tough and value to bring meuntil here,
my dear teachers.
Satu itu Alif, Alif itu Ikhlas
Ikhlas itu perlu diletakkan pada tempat Pertama, Juga Perlu di Utamakan
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Abstract of thesis presented to the Senate of Universiti Putra Malaysia infulfilment of the requirement for the degree of Master of Science
CLASSIFICATION OF SECOND CLASS FILIFORM LEIBNIZ
ALGEBRAS IN DIMENSION NINE
By
FATANAH DERAMAN
August 2013
Chair: Professor Isamiddin S. Rakhimov, PhD
Faculty: Science
This study is centered on part of structural property of Leibniz algebras which
is classification of a subclass of filiform Leibniz algebras over complex field. The
class of filiform Leibniz algebras has been divided into three disjoint classes. The
first two subclasses denoted by FLbn and SLbn arise from naturally graded non-
Lie filiform Leibniz algebras. The third class, TLbn arises from naturally graded
filiform Lie algebras. In particular, the study is focused on the classification prob-
lem of second class filiform Leibniz algebras, SLbn. The method employed in the
classification is to first construct multiplication table of SLb9. Then, adapted
transformation and elementary base change are used to create an isomorphism
criterion. From the isomorphism criterion, the isomorphism classes of algebras
from SLb9 are deduced. Some of these classes are single orbits while others are
parametric family orbits. An invariant algebraic approach is used to describe the
parametric family of orbits. As a result, it is concluded that there are 38 families
of non-isomorphic classes of algebras in SLb9. This comprises 17 single orbits and
21 parametric families of orbits.
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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagaimemenuhi keperluan untuk ijazah Master Sains
PENGELASAN ALJABAR LEIBNIZ FILIFORM KELAS KEDUA
DALAM MATRA SEMBILAN
Oleh
FATANAH DERAMAN
Ogos 2013
Pengerusi: Profesor Isamiddin S. Rakhimov, PhD
Fakulti: Sains
Tesis ini menumpukan kajian kepada salah satu bahagian dalam ciri-ciri struktur
aljabar Leibniz yang merupakan pengelasan bagi subkelas aljabar Leibniz filiform
terhadap medan kompleks. Aljabar Leibniz filiform ini telah dibahagikan kepada
tiga kelas yang tidak bercantum. Dua subkelas yang pertama dilambangi FLbn
dan SLbn yang terbit daripada pengelasan secara asli aljabar Leibniz filiform bukan
Lie. Kelas ketiga, TLbn terbit daripada pengelasan secara asli aljabar Lie filiform.
Secara khususnya kajian ini berfokus kepada pengelasan kelas kedua aljabar Leib-
niz filiform, SLbn. Kaedah yang digunakan dalam kajian pengelasan ini adalah
pertamanya, membina sifir pendaraban dalam SLb9. Kemudian transformasi yang
telah diadaptasi dan pertukaran asas permulaan digunakan untuk menghasilkan
kriterium isomorfisma. Daripada kriterium isomorfisma ini, isomorfisma kelas al-
jabar akan diperoleh dari SLb9. Sebahagian daripada kelas aljabar ini adalah orbit
tunggal, manakala yang lain adalah keluarga orbit berparameter. Hasilnya boleh
dirumuskan sebagai 38 keluarga aljabar tak berisomorfisma di dalam SLb9 yang
terdiri daripada 17 orbit tunggal dan 21 keluarga orbit berparameter.
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ACKNOWLEDGEMENTS
Alhamdulillah, praise to Allah SubhanahuWaTaala for giving me the strength,
guidance and patience to complete this thesis. May the blessing and peace of
Allah be upon Prophet Muhammad Sallalahu Alayhi Wasalam, who was send for
mercy to the world.
I would take this opportunity to express my profound gratitude and deep regards
to my supervisor Prof. Dr. Isamiddin Sattarovich Rakhimov for his exemplary
guidance, monitoring and constant encouragement throughout the course of this
thesis. The blessing, help and guidance given by him time to time shall carry me
a long way in the journey of life. Countless thanks also for his hospitality during
my research attachment visit at Uzbekistan. May Allah bless him for all his kind-
ness. My special thanks also extended to Dr. Siti Hasana Sapar and Dr. Sharifah
Kartini Said Husain for helpful discussions and continuous encouragements. Their
cooperation is indeed appreciated.
My highly appreciation also goes to Prof. Dato’ Dr. Kamel Arifin Mohd Atan, Di-
rector of Institute for Mathematical Research (INSPEM) for his assistance through-
out my study especially through his leadership in providing various facilities and
financial aids, organizing seminars and workshops for the needs of his students.
I also would to express my special thanks to all staffs at School of Graduate Study
UPM, National University of Uzbekistan as well as Institute of Mathematical Re-
search of Uzbekistan especially to Prof. Ayupov, Prof. Omirov, Dr. Abror and
Dr. Jobir Adashev for realizing my dream to be in Uzbekistan from 10 October
until 10 November 2012.
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My colleagues, Abdulhafeez, Azhani Elias, Nur Amalina Azam, Pn. Witriany
Basri and Wan Nurhanan Wan Suhaimi, thank you so much for spending your
time reading and commenting this countless numerical thesis. Also to all my
friends in group research members, Inspem II, Aljabar Friday Seminar Members,
Department of Mathematic UPM and who help me to face and solve the problem
during the course.
Last but not least to my sources of inspiration; my beloved parents, grandparent,
and my siblings for their love and continuous supported me both spiritually and
materially. They never giving up to push and courage me, patiently heard my
stories and cries, always be beside me in good and bad until this completion.
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I certify that a Thesis Examination Committee has met on 22 August 2013
to conduct the final examination of Fatanah Binti Deraman on her thesis enti-tled “Classification of Second Class Nine Dimensional Filiform Leibniz Algbras”in accordance with the Universities and University Colleges Act 1971 and theConstitution of the Universiti Putra Malaysia [P.U.(A) 106] 15 March 1998. TheCommittee recommends that the student be awarded the of Master Of Science.
Members of the Thesis Examination Committee were as follows:
Muhammad Rezal Kamel Ariffin, Ph.D.
Associate ProfessorFaculty of ScienceUniversiti Putra Malaysia(Chairperson)
Mohamad Rushdan Md Said, Ph.D.
Associate ProfessorFaculty of ScienceUniversiti Putra Malaysia(Internal Examiner)
Mat Rofa Ismail, Ph.D.
Associate ProfessorFaculty of ScienceUniversiti Putra Malaysia(Internal Examiner)
Abdul Razak Salleh, Ph.D.
Professor Dato’Fakulti of Science and TechnologyUniversiti Kebangsaan MalaysiaMalaysia(External Examiner)
NORITAH OMAR, Ph.D.
Associate Professor and Deputy DeanSchool of Graduate StudiesUniversiti Putra Malaysia
Date:
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This thesis was submitted to the Senate of Universiti Putra Malaysia and has beenaccepted as fulfilment of the requirement for the degree of of Master of Science.The members of the Supervisory Committee were as follows:
Isamiddin S. Rakhimov, PhD
ProfessorFaculty of ScienceUniversiti Putra Malaysia(Chairperson)
Siti Hasana Sapar, PhD
Senior LecturerFaculty of ScienceUniversiti Putra Malaysia(Member)
Sharifah Kartini Said Husain, PhD
LecturerFaculty of ScienceUniversiti Putra Malaysia(Member)
BUJANG KIM HUAT, PhD
Professor and DeanSchool of Graduate StudiesUniversiti Putra Malaysia
Date:
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DECLARATION
I declare that the thesis is my original work except for quotations and citationswhich have been duly acknowledged. I also declare that it has not been previ-ously, and is not concurrently, submitted for any other degree at Universiti PutraMalaysia or at any other institution.
FATANAH DERAMAN
Date: 22 August 2013
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TABLE OF CONTENTS
Page
DEDICATIONS ii
ABSTRACT iii
ABSTRAK iv
ACKNOWLEDGEMENTS v
APPROVAL vii
DECLARATION ix
LIST OF FIGURES xii
LIST OF ABBREVIATIONS xiii
CHAPTER
1 INTRODUCTION 11.1 Literature review 11.2 Objectives 51.3 Outline of chapters 5
2 BASIC CONCEPTS AND METHODOLOGY 62.1 Basic concepts 62.2 Methodology 11
3 ISOMORPHISM CRITERION 213.1 The isomorphism criterion 21
4 RESULTS AND DISCUSSIONS : β3 6= 0 324.1 Isomorphism criteria 324.2 The similarity of isomorphism criteria 344.3 Disjoint subsets 414.4 Description of parametric family orbits in invariant form 424.5 Description of single orbits in non-invariants form 53
5 RESULTS AND DISCUSSIONS : β3 = 0 615.1 Isomorphism criteria and disjoint subsets 615.2 Description of parametric family orbits in invariants form 655.3 Description of single orbits in non-invariant form 74
6 CONCLUSION AND FUTURE WORK 826.1 Summary 826.2 Recommendations for Future Research 88
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LIST OF ABBREVIATIONS
Lbn The class of n dimensional filiform Leibniz algebras
LNn The class of n dimensional nilpotent Leibniz algebras
FLbn First class of n dimensional complex filiform Leibniz al-
gebras
SLbn Second class of n dimensional complex filiform Leibniz
algebras
TLbn Third class of n dimensional complex filiform Leibniz al-
gebras
GLn(K) Group of n× n nonsingular matrices over K
Algn(K) Algebraic variety of n dimensional algebra structures over
a field K
Gad Subgroup of all adapted transformations of GLn(K)
NGFi i class of naturally graded filiform Leibniz algebras
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CHAPTER 1
INTRODUCTION
This thesis presents the results on classification problems of second class of fili-
form Leibniz algebras in dimension nine. This class arises from naturally graded
non-Lie filiform Leibniz algebras. In general, the classification problem of algebras
is an age long problem which cannot be overemphasized. Therefore, we have to
consider some subclasses under some conditions. Nilpotency and filiformity are
some of such kind conditions.
This chapter contains preface to get a hold of literature review, research objec-
tives and sketch of the thesis chapters. The methods used in this study gathered
tools from different known concepts such as group action, linear transformation,
elementary base change, just to mention but a few.
This chapter also contains an engrossing and fast-paced historical review combine
with current study which brought an answer for the readers who are anxious about
the present research.
1.1 Literature review
According to Helgason (1994), classifications and simplifications of transformation
groups have been initiated by Sophus Lie. He also introduced a class of non-
commutative and non-associative algebras which is known now as the class of Lie
algebra. The extensive studies on the cohomological problems of algebras by Jean-
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Louis Loday in 1990s, leads to a generalization of the Lie algebras called Leibniz
algebras (Loday and Pirashvili, 1993). The Leibniz algebras have been found to
be closely related to Lie algebras. As for the classification problem of finite dimen-
sional Lie algebras, it can be divided into three parts. They are the classification
of nilpotent Lie algebras, the description of solvable Lie algebras with given nil-
radical and the description of Lie algebras with given radical. The classification
of nilpotent complex Lie algebras is only known in dimension up to seven. The
description of solvable Lie algebras has been reduced to the description of orbits
of certain unipotent linear groups. The description of solvable Lie algebras has
been reduced to the description of semisimple subalgebras whose classification has
been known in the work of Cartan-Killing. In dimension six there are several clas-
sification results over a field of characteristic zero and an arbitrary field has been
given by Morozov (1958), Shedler (1964), respectively. Romdhani (1989) studied
on dimension seven. All these results were obtained using a variety of isomorphism
invariants. There are some other results on classification of subclasses of nilpotent
Lie algebras such as Scheuneman (1967), Gauger (1973), Galitski and Timashev
(1999) and Gomez et al. (1998) (filiform Lie algebras) and others.
In recent year, there has been an increasing interest on investigation of Leibniz
algebras problems. Albeverio et al. (2005) describe the irreducible components of
the nilpotent complex Leibniz algebras varieties of dimension less than five. They
also construct degenerations between one parametric families of nilpotent Leib-
niz algebras and study the rigidity of these families. In the same years, Omirov
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(2005) studied analogues of zero filiform and filiform Leibniz algebras in the fi-
nite dimensional case. The Leibniz superalgebras are generalizations of Leibniz
algebras. This area can be investigated by applying restrictions on their charac-
teristic sequence (Ayupov et al., 2009). By using Mathematica software, Casas
et al. (2009) develop a computer program to check the Leibniz identity. Cabezas
et al. (2011) present the classification of a subclass of naturally graded Leibniz
algebras. Meanwhile, Casas et al. (2013) extend the classification of solvable Lie
algebras with naturally graded filiform Lie algebra to the case of Leibniz algebras.
Namely, the classification of solvable Leibniz algebras whose nilradical is naturally
graded filiform Leibniz algebras.
The structure theory of Leibniz algebras is studied through derivations, classifi-
cation, Cartan subalgebras, weight spaces, solvability and etc. The derivations
of naturally graded Leibniz algebras are described by Omirov (2005). Albeverio
et al. (2006b) give the classification of nilpotent Leibniz algebras up to dimension
four. A further study on the classification of low dimensional filiform Leibniz al-
gebras over a field of p-adic numbers is done by Ayupov and Kurbanbaev (2010).
Rikhsiboev and Rakhimov (2011) completed the classification of three dimensional
complex Leibniz algebras. Besides that, the details on properties of Cartan sub-
algebras, weight space and solvability of finite dimensional Leibniz algebras are
discussed in Albeverio et al. (2006a) and Zargeh (2012).
The notion of filiformity for Leibniz algebras was introduced by Ayupov and
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Omirov (2001). In the same paper the authors deal with the class of Leibniz
algebras arising from naturally graded complex non-Lie filiform Leibniz algebras.
The authors showed that this class is split into two subclasses. Later, Gomez and
Omirov (2006) gave the isomorphism criteria for these two classes and they men-
tioned an existence of one more class of filiform Leibniz algebras which arises from
naturally graded filiform Lie algebras. The third class of filiform Leibniz algebras
has been treated by Omirov and Rakhimov (2009). They gave an adapted basis
with respect to this class and were written in simple table form. As for the first two
classes Rakhimov and Bekbaev (2010), suggested an approach based on algebraic
invariant to classify these two classes in fixed dimensional case. This classification
approach uses the notion of group action. Furthermore, it was stated that the
results can also be used in geometric classification of filiform Leibniz algebras.
Sozan et al. (2010) gave the classification of nine dimensional first class of complex
filifom Leibniz algebras. The classification up to dimension six of the third class
of complex filiform Leibniz algebras has been given by Hassan (2010). The clas-
sification problem of lower dimensional cases of the first and second classes have
been studied by Rakhimov and Said Husain (2011a,b). The aim of this thesis is
to give a complete classification of the second class of filiform Leibniz algebras in
dimension nine.
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1.2 Objectives
The main objectives of this research are:
1. To give a multiplication table of nine dimensional second class of complex fili-
form Leibniz algebras (SLb9).
2. To create isomorphism criterion for SLb9.
3. To list the isomorphism classes and their respective orbits.
4. To give invariants in parametric orbits case.
1.3 Outline of chapters
This thesis contains six chapters. It is divided into three main sections which
are introduction, results and conclusion. In Chapter 1, the historical background
and objectives are discussed. Meanwhile, Chapter 2 focuses on the basic concepts
and the methodology employed in this research. The main results are presented
in Chapter 3, Chapter 4 and Chapter 5. Basically, Chapter 3 focuses on isomor-
phism criterion. In Chapter 4 and Chapter 5 the isomorphism classes and their
respective invariants are given. Finally, Chapter 6 summarizes the results and
gives recommendations for further research.
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REFERENCES
Albeverio, S., Ayupov, S. A., and Omirov, B. A. (2005). On nilpotent and simpleLeibniz algebras. Communications in Algebra, 33(1):156–172.
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Albeverio, S., Omirov, B. A., and Rakhimov, I. S. (2006b). Classification of 4-dimensional nilpotent complex Leibniz algebras. Extrac Math., 21(3):197–210.
Ayupov, S. A. and Kurbanbaev, T. (2010). The classification of 4-dimensionalp-adic filiform Leibniz algebras. J. Pure Appl. Math., 1(2):155–162.
Ayupov, S. A. and Omirov, B. A. (2001). On some classes of nilpotent Leibnizalgebras. Siberian Mathematical Journal, 42(1):15–24.
Ayupov, S. A., Omirov, B. A., and Khudoyberdiyev, A. K. (2009). The classifi-cation of filiform Leibniz superalgebras of nilindex n + m. Acta Mathematica
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